Parameter-Replacement Functions for Stability-Guaranteed Variable Digital Filters

Digital filters are the most basic and most useful signal-processing systems that are used to pass through some frequency components in a signal and eliminate others. During filtering operations, a digital filter being applied usually needs to change its frequency-domain characteristic (frequency response) as quickly as possible. To achieve this objective, a digital filter must have the capability to change its frequency response in a real-time manner, and such a digital filter is called variable digital filter (VDF). VDFs have changeable coefficients, and thus variable frequency responses. To get a new frequency response, a VDF needs to change its coefficients. On the other hand, changing the coefficients of a recursive VDF may cause a big problem, which is instability. That is, the new coefficients may make the recursive VDF become unstable. Therefore, it is mandatory to ensure the filter stability when the filter coefficients are changed. So far, a technique for ensuring the stability of a recursive VDF has been revealed. The tactic is based on coefficient replacements. Specifically, this technique uses a function to replace the denominator coefficients of the transfer function with a kind of composite functions, and those composite functions involve a few new parameters. In the VDF design, the new parameters involved in the composite functions are sought by minimizing a specific error measure (say, mean-squared error). The key point is that no matter what values the new parameters take, the composite functions can always ensure the stability of a recursive VDF. To replace the denominator coefficients with composite functions, a special function called parameter-replacement (PR) function is employed. In this paper, a wide variety of PR functions are presented.